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Estimating location parameters of two exponential distributions with ordered scale parameters

Lakshmi Kanta Patra, Constantinos Petropoulos, Shrajal Bajpai, Naresh Garg

Abstract

In the usual statistical inference problem, we estimate an unknown parameter of a statistical model using the information in the random sample. A priori information about the parameter is also known in several real-life situations. One such information is order restriction between the parameters. This prior formation improves the estimation quality. In this paper, we deal with the component-wise estimation of location parameters of two exponential distributions studied with ordered scale parameters under a bowl-shaped affine invariant loss function and generalized Pitman closeness criterion. We have shown that several benchmark estimators, such as maximum likelihood estimators (MLE), uniformly minimum variance unbiased estimators (UMVUE), and best affine equivariant estimators (BAEE), are inadmissible. We have given sufficient conditions under which the dominating estimators are derived. Under the generalized Pitman closeness criterion, a Stein-type improved estimator is proposed. As an application, we have considered special sampling schemes such as type-II censoring, progressive type-II censoring, and record values. Finally, we perform a simulation study to compare the risk performance of the improved estimators

Estimating location parameters of two exponential distributions with ordered scale parameters

Abstract

In the usual statistical inference problem, we estimate an unknown parameter of a statistical model using the information in the random sample. A priori information about the parameter is also known in several real-life situations. One such information is order restriction between the parameters. This prior formation improves the estimation quality. In this paper, we deal with the component-wise estimation of location parameters of two exponential distributions studied with ordered scale parameters under a bowl-shaped affine invariant loss function and generalized Pitman closeness criterion. We have shown that several benchmark estimators, such as maximum likelihood estimators (MLE), uniformly minimum variance unbiased estimators (UMVUE), and best affine equivariant estimators (BAEE), are inadmissible. We have given sufficient conditions under which the dominating estimators are derived. Under the generalized Pitman closeness criterion, a Stein-type improved estimator is proposed. As an application, we have considered special sampling schemes such as type-II censoring, progressive type-II censoring, and record values. Finally, we perform a simulation study to compare the risk performance of the improved estimators

Paper Structure

This paper contains 16 sections, 21 theorems, 115 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Improved estimation of $\mu_1$
  3. Improvement over the BAEE of $\mu_1$
  4. A class of improved estimator for $\mu_1$
  5. Improved estimation of $\mu_2$
  6. Improvement over the BAEE of $\mu_2$
  7. A class of improved estimators for $\mu_2$
  8. Improved estimation under Generalized Pitman Closeness
  9. Improved estimation of $\mu_{1}$
  10. Improved estimation of $\mu_{2}$
  11. Application to special sampling schemes
  12. Type II censoring
  13. Progressive type II censoring
  14. Record values
  15. Simulation Study
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $U_1 = \frac{X_{1(1)} - \mu_1}{\sigma_1}$, which follow an exponential $Exp\left(\frac{1}{n_1}\right)$ distribution and $V_1=\frac{T_1}{\sigma_1}$, which follow a $Gamma(n_1-1,1)$ distribution. Then, under an arbitrary bowl-shaped loss function $L(t)$, the (unrestricted) BAEE $\delta_{01}(\under

Theorems & Definitions (42)

  • Theorem 2.1
  • Example 2.1
  • Example 2.2
  • Remark 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Example 2.3
  • Example 2.4
  • Theorem 2.4
  • Corollary 2.5
  • ...and 32 more